Representation of Saturation in Stability Studies

Transcription

1 Inteestng summay: Repesentaton of atuaton n tabty tues Kunu wtes (pg ) that A goous teatment of synchonous machne pefomance ncung satuaton effects s a fute execse. Any pactca metho of accountng fo satuaton effects must be base on sem-heustc easonng an ucousy chosen appoxmatons, wth ue conseaton to smpcty of moe stuctue, ata aaabty, an accuacy of esuts. ome assumptons (see Kunu pg. 2-3) :. eakage nuctances ae nepenent of satuaton snce the path of the eakage fux s many n the a. Theefoe we may confne ou anayss of satuaton to the mutua nuctances, epesente by A an A. 2. The eakage fuxes o not contbute to the on satuaton. Ths s easonabe because these fuxes ae sma (snce the paths ae many n a, an a has hgh pemeabty), an the paths conce wth that of the man fux fo ony a sma pat of ts path. o we may etemne satuaton of the nuctances as a functon of A an A. 3. The satuaton eatonshp between the esutant a-gap fux an the mmf une oae contons s the same as une no-oa contons. Ths aows the satuaton chaactestcs to be epesente by the open-ccut satuaton cue, whch s usuay the ony satuaton ata eay aaabe. An atona assumpton that s sometmes mae s that A oes not satuate, smpy because the uaatue axs fux s usuay ute sma n compason to the ect axs fux ue to the effect of the man fe wnng. Ths assumpton s ute goo fo saent poe machnes but not so goo fo oun-oto machnes.

2 Reca, fom ou euaent ccut (shown beow), that A =( + + ) A. [ ] A A[ ] [ ] A A ect-axs euaent ccut: The aboe s the same as g. 4.5 n you text efne the foowng tems: agnetzaton cuent: =( + + ) A = A axmum pe-unt fux nkage wthout satuaton: AT : cuent that wou pouce A f no satuaton effects : cuent that pouces A wth satuaton effects : ux nkage esutng fom f no satuaton effects efne A as the nuctance coesponng to the a-gap ne. It s the nuctance when s sma,.e., t s the non-satuate nuctance. The magnetzaton cue appeas as n the foowng fgue: 2

3 A AT om the fgue, we can wte that: A A But fom the a-gap ne euaton, of ths eaton nto the peous one yes: A A A, an substtuton efne K = /,. K s the facton of the satuate cuent necessay to achee the same fux nkage wth no satuaton (.e., on the agap ne). Ceay, <K <, whee K cose to ncates a hghy satuate ; K cose to ncates a non-satuate. o hee we nee to ecognze a ey mpotant featue: K epens on the satuaton ee whch epens on λ A. ubsttuton of K nto (*) esuts n: A K A o K s a facto that we use to account fo the ffeence between the magnetzaton cue an the a-gap ne. Obseaton: We ae tyng to use K to compute λ A, yet K epens on λ A, that s, λ A =K (λ A ) A. Ths s why Kunu wote, A goous teatment of satuaton effects s a fute execse (see p. of these notes). (*) 3

4 o how mght we etemne K? Obsee: whee = -. K o eauaton of K eues eauaton of, an ou pobem s now to get. Note fom g. beow that gows exponentay age wth A - AT. ength of ne s AT g. o we eason that a goo appoxmaton to s gen by o that K A e A e B ( A AT 4 ( B A ) AT )

5 Now t s cea fom the aboe that K s a functon of A,.e., o that the mutua fux s gen by o how o we use t? Assume that we hae aues fo,,,,, an. Then the steps fo ncung satuaton ae:. Use the auxay euatons to obtan the unsatuate aues of A an A : whee A K K ( A ) K (A) A A A A A 2. o a saent poe machne, et = A. o a oun-oto machne, et =st{ 2 A + 2 A } 3. Check f > AT. If not, use the unsatuate aues. If so, pocee to step Obtan cuents fom 4.24, shown beow: 5

6 6 (4.24 ) 5. Compute the magnetzng cuent as 6. Compute K accong to: B K A e AT ) ( 7. Upate A an A accong to a. Repace A wth A K A, an then compute: A A b. If saent poe, then A = A (.e., no change), but f ounoto, then epace A wth A K A, an then compute: A A A A

7 7 An then you can use the upate aues of A an A n the foowng to pefom a numeca ntegaton an get the next tme step A (4.26) A (4.28) A (4.29) A (4.3) A (4.3) A (4.3 ) A A m T 3 3 (4.33) (4.2) Reaton to nput ata to most commeca stabty pogams: The nput euements fo chaactezng geneato satuaton fo most commeca-gae stabty pogams ae n tems of a paamete cae, efne by whee = - as befoe. Reca that

8 8 The eaton between an K s ee fom the beow: K The specfc ata enty nto most pogams (ncung P/E) s (.): aue of when open ccut temna otage s.pu (.2): aue of when open ccut temna otage s.2pu Note that (.2) shou aways be age than (.). In the abo Canyon ata, (.) s.769 an (.2) s.4. The coesponng aues of K ae.9286 an.792, espectey. The fgue beow ustates teatment of satuaton fom one commeca gae stabty pogam. K

9 g. 2 9

10 na note on satuaton. ecton eeops a moe whee satuaton s negecte. uch a moe s usefu fo neaze anayss. The appoach s smpe ust substtute the auxay euatons,.e., the expessons fo λ A an λ A,.e., A A nto the state euatons: A (4.26) A (4.28) A (4.29) A (4.3) A (4.3) A (4.3 ) A A m T 3 3 (4.33) (4.2) You book oes ths fo a of the aboe, except, of couse, fo the euaton coesponng to the -wnng (the state euaton fo λ ). o we w o t fo that one. We aso must o t fo (4.3), (4.3), an (4.33) snce these euatons contan λ A whch contans λ.

11 st, we hane the state euaton fo λ. 2 Now fo (4.3): A An now fo (4.3): A An fnay fo (4.33):

12 T m A A 3 3 T m 3 3 T m The aboe state euatons nees to be ncue to the state euatons gen n the book, e. (4.38), whch ae poe beow: Tm HOEWORK: Repeat Exampe 4.3 n the text but ncue -cct. Comments:. Thee s a ey goo teatment of satuaton n IEEE t -22, IEEE ue fo ynchonous eneato oeng Pactces an Appcatons n Powe ystem tabty Anayses, see chapte 6. Inee, ths s a ey goo stana to compement ou couse, an I encouage you to ownoa, pnt, an ncue n you foe. 2

13 2. Note the pesence of an on the ght-han-se. Reca that =P abc an so an come fom the phase otages a, b, an c. nce the phase otages ae affecte by the oa cuents, so ae an. o, we nee to epesent the oa n oe to compete the moe. Ths s the subect of secton